Fully reliable error control for evolutionary problems

TL;DR

This paper develops and tests functional-type a posteriori error estimates for time-dependent evolutionary problems, comparing time-marching and space-time solution reconstructions, and demonstrating their effectiveness through extensive numerical experiments.

Contribution

It introduces new algorithms for error estimation in evolutionary problems, including a global minimization approach and application to space-time methods, with validated numerical results.

Findings

The global minimization algorithm effectively reduces error bounds.

Functional error estimates are efficient for space-time approaches.

Numerical tests confirm the reliability of the proposed error control methods.

Abstract

This work is focused on the application of functional-type a posteriori error estimates and corresponding indicators to a class of time-dependent problems. We consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results. This paper examines two different methods of approximate solution reconstruction for evolutionary models, i.e., a time-marching technique and a space-time approach. The first part of the study presents an algorithm for global minimization of the majorant on each of discretization time-cylinders (time-slabs), the effectiveness of this algorithm is confirmed by extensive numerical tests. In the second part of the publication, the application of functional error estimates is discussed with respect to a space-time approach. It is followed by a set of extensive numerical tests that demonstrate the efficiency of proposed error control method.